Computational Challenges of Inverse Problems
Thursday, February 22, 2018
3:30PM – 5PM
Inverse problems are omnipresent in many scientific fields such as systems biology, engineering, medical imaging, and geophysics. The main challenges toward obtaining meaningful real-time solutions to large, data-intensive inverse problems are ill-posedness of the problem, large parameter dimensions, and/or complex model constraints. This talk discusses computational challenges of inverse problems by exploiting a combination of tools from applied linear algebra, parameter estimation and optimization, and statistics. For instance, for large scale ill-posed inverse problems, approximate solutions are computed using a regularization method that solves a nearby well-posed problem. Oftentimes, the selection of a proper regularization parameter is the most critical and computationally intensive task and may hinder real-time computations of the solution. We present a new framework for solving ill-posed inverse problems by computing optimal regularized inverse matrices. We further discuss randomized Newton and randomized quasi-Newton approaches to efficiently solve large linear least-squares problems, where the very large data sets present a significant computational burden (e.g., the size may exceed computer memory or data are collected in real-time). In this framework, randomness is introduced as a means to overcome computational limitations, and probability distributions that can exploit structure and/or sparsity are considered. We will present numerical examples, from deblurring, tomography, and machine learning to illustrate the challenges and our proposed methods.
Matthias (Tia) Chung is an Assistant Professor in the Department of Mathematics at Virginia Tech and member of the Computational Modeling and Data Analytics division in the Academy of Integrated Science. He joined the Virginia Tech in 2012, holds a Dipl. math. (Master of Science equivalent) from the University of Hamburg, Germany, and a Dr. rer. nat. (Ph.D. equivalent degree) in Computational Mathematics from the University of Lübeck, Germany. Before joining Virginia Tech, he was a Post-Doctoral Fellow at Emory University and Assistant Professor at Texas State University. Matthias Chung is an active member of the Society for Industrial and Applied Mathematics (SIAM) and its CSE, UQ, IS, and LA activity groups. Matthias Chung’s research concerns various forms of cross-disciplinary inverse problems. Driven by its application, he and his lab develops and analyzes efficient numerical methods for inverse problems. Applications of interest include, but are not limited to, systems biology, medical and geophysical imaging, and dynamical systems. Challenges such as ill-posedness, large-scale, and uncertainty estimates are addressed by utilizing tools from and developing methods for regularization, randomized methods, stochastic learning, Bayesian inversion, and optimization. Research project are supported by NSF, NIH, and USDA.
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